Page 79 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Hint Solve by equating corresponding entries on the two sides.

Prove:
24.

(a) part (b) of Theorem 1.4.1

(b) part (i) of Theorem 1.4.1

(c) part (m) of Theorem 1.4.1

Apply parts (d) and (m) of Theorem 1.4.1 to the matrices A, B, and to derive the result in part (f).
25.

Prove Theorem 1.4.2.
26.

Consider the laws of exponents and .
27.

(a) Show that if A is any square matrix, then these laws are valid for all nonnegative integer values of r and s.

(b) Show that if A is invertible, then these laws hold for all negative integer values of r and s.

Show that if A is invertible and k is any nonzero scalar, then for all integer values of n.
28.

29. , then .
(a) Show that if A is invertible and

(b) Explain why part (a) and Example 3 do not contradict one another.

Prove part (c) of Theorem 1.4.1. , and C is . The th entry on the left side is
30.

Hint Assume that A is , B is

and the th entry on the right side is

. Verify that .

Let A and B be square matrices with the same size.
31.